F.4 Maths

a)
Let f(x) = x^1024 + 1

When f(x) is divided by x + 1, the remainder
= f(-1)
= (-1)^1024 + 1
= 1 + 1
= 2


b)
Put x = 11 :
When 11^1024 + 1 is divided by 12, the remainder = 2
When 11^1024 is divided by 12, the remainder = 1
It will be one month after January.
Hence, after 11^1024 month, it will be February.

For (b),
put x = 11,
The remainder of 11^1024 + 1 divided by 12 = 2 (By (a))
That means the (11^1024 + 1)nd month after january is March
We can then concluded that (11^1024)st month after january is February

2012-03-21 22:53:57 補充：
You may take the following step as Example,
Give (x^1024 +1) = (x+1)Q(x) + 2 (By (a)) For Q(x) is a 1023 degree function.
Put x = 11,
(11^1024 +1) = 12Q(11) + 2 For Q(11) is certain constant
11^1024 = 12Q(11) + 1
Therefore, after 11^1024 months,
the month = the month after January = February

Let f(x) be x^(1024)+1.

By remainder thm,when f(x) is divided by x+1, the remainder is f(-1)=(-1)^(1024) +1=1+1=2